diff --git a/engine-proto/ConstructionViewer.jl b/engine-proto/ConstructionViewer.jl
index 7cb0450..bdd35fd 100644
--- a/engine-proto/ConstructionViewer.jl
+++ b/engine-proto/ConstructionViewer.jl
@@ -123,12 +123,24 @@ mutable struct ConstructionViewer
   end
 end
 
+mprod(v, w) =
+  v[1]*w[1] + v[2]*w[2] + v[3]*w[3] + v[4]*w[4] - v[5]*w[5]
+
 function display!(viewer::ConstructionViewer, elements::Matrix)
   # load elements
-  elements_full = [
-    [0; elt; fill(0, 26)]
-    for elt in eachcol(elements)
-  ]
+  elements_full = []
+  for elt in eachcol(elements)
+    if mprod(elt, elt) < 0.5
+      elt_full = [0; elt; fill(0, 26)]
+    else
+      # `elt` is a spacelike vector, representing a generalized sphere, so we
+      # take its Hodge dual before passing it to Ganja.js. the dual represents
+      # the same generalized sphere, but Ganja.js only displays planes when
+      # they're represented by vectors in grade 4 rather than grade 1
+      elt_full = [fill(0, 26); -elt[5]; -elt[4]; elt[3]; -elt[2]; elt[1]; 0]
+    end
+    push!(elements_full, elt_full)
+  end
   @js viewer.win elements = $elements_full.map((elt) -> @new CGA3(elt))
   
   # generate palette. this is Gadfly's `default_discrete_colors` palette,
diff --git a/engine-proto/ganja-test/ganja-test.html b/engine-proto/ganja-test/ganja-test.html
index d6efc90..0207dcc 100644
--- a/engine-proto/ganja-test/ganja-test.html
+++ b/engine-proto/ganja-test/ganja-test.html
@@ -28,6 +28,25 @@
     CGA3.inline(() => Math.sqrt(0.5)*(-1e1 - 1e2 + 1e3 + 1e5))(),
     CGA3.inline(() => -Math.sqrt(3)*1e4 + Math.sqrt(2)*1e5)()
   ];
+  /*
+    these blocks of commented-out code can be used to confirm that a spacelike
+    vector and its Hodge dual represent the same generalized sphere
+  */
+  /*let elements = [
+    CGA3.inline(() => Math.sqrt(0.5)*!( 1e1 + 1e2 + 1e3 + 1e5))(),
+    CGA3.inline(() => Math.sqrt(0.5)*!( 1e1 - 1e2 - 1e3 + 1e5))(),
+    CGA3.inline(() => Math.sqrt(0.5)*!(-1e1 + 1e2 - 1e3 + 1e5))(),
+    CGA3.inline(() => Math.sqrt(0.5)*!(-1e1 - 1e2 + 1e3 + 1e5))(),
+    CGA3.inline(() => !(-Math.sqrt(3)*1e4 + Math.sqrt(2)*1e5))()
+  ];*/
+  /*let elements = [
+    CGA3.inline(() =>  1e1 + 1e5)(),
+    CGA3.inline(() =>  1e2 + 1e5)(),
+    CGA3.inline(() =>  1e3 + 1e5)(),
+    CGA3.inline(() => -1e4 + 1e5)(),
+    CGA3.inline(() => Math.sqrt(0.5)*(1e1 + 1e2 + 1e3 + 1e5))(),
+    CGA3.inline(() => Math.sqrt(0.5)*!(1e1 + 1e2 + 1e3 - 0.01e4 + 1e5))()
+  ];*/
   
   // set up palette
   var colorIndex;
@@ -66,6 +85,7 @@
       
       // de-noise
       for (let k = 6; k < elements[n].length; ++k) {
+      /*for (let k = 0; k < 26; ++k) {*/
         elements[n][k] = 0;
       }
     }